1. Section 1.2–1.3
A Catalog of Essential Functions
The Limit of a Function
V63.0121.027, Calculus I
September 10, 2009
Announcements
Syllabus is on the common Blackboard
Office Hours MTWR 3–4pm
Read Sections 1.1–1.3 of the textbook this week.
. . . . . .
2. Outline
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .
3. Classes of Functions
linear functions, defined by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions
. . . . . .
4. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
. . . . . .
5. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.
. . . . . .
6. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.
Answer
If x is in miles and f(x) in dollars,
f(x) = 2.5 + 2x
. . . . . .
8. Quadratic functions
These take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0, downward
if a < 0.
. . . . . .
9. Cubic functions
These take the form
f(x) = ax3 + bx2 + cx + d
. . . . . .
10. Other power functions
Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.
. . . . . .
11. Rational functions
Definition
A rational function is a quotient of polynomials.
Example
x 3 (x + 3 )
The function f(x) = is rational.
(x + 2)(x − 1)
. . . . . .
12. Trigonometric Functions
Sine and cosine
Tangent and cotangent
Secant and cosecant
. . . . . .
13. Exponential and Logarithmic functions
exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example
f(x) = log2 (x))
. . . . . .
14. Outline
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .
15. Transformations of Functions
Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
. . . . . .
16. Transformations of Functions
Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
Observe that if the fiddling occurs within the function, a
transformation is applied on the x-axis. After the function, to the
y-axis.
. . . . . .
17. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
18. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
19. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
20. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
21. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right
y = f(x + c), shift the graph of y = f(x) a distance c units to
the left
. . . . . .
22. Outline
Classes of Functions
Linear functions
Quadratic functions
Cubic functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
Limits
Heuristics
Errors and tolerances
Examples
Pathologies
. . . . . .
24. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
. . . . . .
25. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the
same.
. . . . . .
26. Decomposing
Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?
Solution √
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f.
To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
. . . . . .
30. Zeno’s Paradox
That which is in
locomotion must
arrive at the
half-way stage
before it arrives at
the goal.
(Aristotle Physics VI:9,
239b10)
. . . . . .
31. Heuristic Definition of a Limit
Definition
We write
lim f(x) = L
x→a
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) but not equal to a.
. . . . . .
32. The error-tolerance game
A game between two players to decide if a limit lim f(x) exists.
x→a
Player 1: Choose L to be the limit.
Player 2: Propose an “error” level around L.
Player 1: Choose a “tolerance” level around a so that
x-points within that tolerance level are taken to y-values
within the error level.
If Player 1 can always win, lim f(x) = L.
x→a
. . . . . .
35. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
36. The error-tolerance game
T
. his tolerance is too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
37. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
38. The error-tolerance game
S
. till too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
39. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
40. The error-tolerance game
T
. his looks good
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
41. The error-tolerance game
S
. o does this
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
. . . . . .
42. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .
43. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical)
strip must also be inside the green (horizontal) strip.
If Player 2 shrinks the error, Player 1 can still win.
. . . . . .
45. Example
Find lim x2 if it exists.
x→0
Solution
I claim the limit is zero.
. . . . . .
46. Example
Find lim x2 if it exists.
x→0
Solution
I claim the limit is zero.
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufficiently close to zero.
. . . . . .
47. Example
Find lim x2 if it exists.
x→0
Solution
I claim the limit is zero.
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufficiently close to zero.
If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round.
. . . . . .
48. Example
Find lim x2 if it exists.
x→0
Solution
I claim the limit is zero.
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufficiently close to zero.
If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round.
What should the tolerance be if the error is 0.0001?
. . . . . .
49. Example
Find lim x2 if it exists.
x→0
Solution
I claim the limit is zero.
If the error level is 0.01, I need to guarantee that
−0.01 < x2 < 0.01 for all x sufficiently close to zero.
If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round.
What should the tolerance be if the error is 0.0001?
By setting tolerance equal to the square root of the error, we can
guarantee to be within any error.
. . . . . .
50. Example
|x|
Find lim if it exists.
x→0 x
. . . . . .
51. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
. . . . . .
59. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
. . . . . .
60. The error-tolerance game
y
.
.
Part of graph in-
side blue is not . .
1
inside green
. x
.
. 1.
−
These are the only good choices; the limit does not exist.
. . . . . .
61. One-sided limits
Definition
We write
lim f(x) = L
x→a+
and say
“the limit of f(x), as x approaches a from the right, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) and greater than a.
. . . . . .
62. One-sided limits
Definition
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the left, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to
L as we like) by taking x to be sufficiently close to a (on either
side of a) and less than a.
. . . . . .
63. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−
. . . . . .
64. Example
1
Find lim if it exists.
x→0+ x
. . . . . .
71. The error-tolerance game
y
.
.
The limit does not exist
because the function is
unbounded near 0
.? .
L
. x
.
0
.
. . . . . .
72. Example
1
Find lim if it exists.
x→0+ x
Solution
The limit does not exist because the function is unbounded near
0. Next week we will understand the statement that
1
lim = +∞
x→0+ x
. . . . . .
73. Weird, wild stuff
Example (π )
Find lim sin if it exists.
x→0 x
. . . . . .
74. Weird, wild stuff
Example (π )
Find lim sin if it exists.
x→0 x
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
75. Weird, wild stuff
Example (π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
76. Weird, wild stuff
Example (π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
1
f(x) = 1 when x = for any integer k
2k + 1/2
f(x) = −1 when x =
. . . . . .
77. Weird, wild stuff
Example (π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
1
f(x) = 1 when x = for any integer k
2k + 1/2
1
f(x) = −1 when x = for any integer k
2k − 1/2
. . . . . .
78. Weird, wild stuff continued
Here is a graph of the function:
y
.
. .
1
. x
.
. 1.
−
There are infinitely many points arbitrarily close to zero where
f(x) is 0, or 1, or −1. So the limit cannot exist.
. . . . . .
79. What could go wrong?
Summary of Limit Pathologies
How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a
. . . . . .
80. Meet the Mathematician: Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis,
number theory
created the definition of
limit we use today but
didn’t understand it
. . . . . .
81. Precise Definition of a Limit
No, this is not going to be on the test
Let f be a function defined on an some open interval that
contains the number a, except possibly at a itself. Then we say
that the limit of f(x) as x approaches a is L, and we write
lim f(x) = L,
x→a
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ , then |f(x) − L| < ε.
. . . . . .